Exact coherent solutions for the filtered large scale motions in turbulent Couette flow
Résumé
We investigate the nature of large-scale motions (LSM) in turbulent Couette flow. It is first shown that large-scale motions are self-sustained by verifying that they survive even when active processes at smaller scales are artificially quenched (see figure 1) in over-damped large eddy simulations as in [1, 2]. We then investigate the nature of the self-sustained large-scale motions by looking for exact solutions of the filtered, possibly over-damped, (LES) equations. The edge state of coherent large-scale motions is computed by edge-tracking and is found to be a non-trivial exact steady solution of the filtered equations. Newton-Krylov iterations are used to prove convergence of the steady solution and to perform parameter-continuations. By continuation to higher values of the over-damping it is shown that the lower branch edge state is connected to an upper branch solution via a saddle-node bifurcation. We have been able to compute upper branch solutions of the filtered equations at Reynolds numbers up to Re = 2150 using specific paths in the Reynolds number-overdamping parameter plane (see figure 2). These solutions are then continued to Navier-Stokes solutions by reducing to zero the residual stress in the LES. It is finally shown that the steady large-scale solutions of the filtered equations motions can be connected by continuation to the Nagata-Clever-Busse-Waleffe solutions of the Navier-Stokes equations. Financial support from PRES Université de Toulouse and Région Midi-Pyrénées is kindly acknowledged.
References
[1] Y. Hwang and C. Cossu. Self-sustained process at large scales in turbulent channel flow. Phys. Rev. Lett., 105(4):044505, 2010.
[2] Y. Hwang and C. Cossu. Self-sustained processes in the logarithmic layer of turbulent channel flows. Phys. Fluids,